Mathematical Tools for Neuroscience: A Geometric Approach

The book explains how the brain is adept at processing biological objects by exploiting the geometric properties of such objects. It introduces the concept of a manifold, which extends the idea of a surface to many dimensions, and discusses how to use tangent spaces to investigate the local linear behavior of manifolds. Examples of the geometric approach to neuroscience include the analysis of color and spatial vision measurements and the control of eye and arm movements. The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach.

Format: Hardback
Length: 162 pages
Publication date: 22 April 2022
Publisher: Springer Nature Switzerland AG

This comprehensive book offers a concise yet accessible introduction to a diverse range of mathematical concepts that have proven invaluable in understanding the complexities of the brain and behavior. By meticulously recording the eye movements of individuals observing a riverside scene, one can observe a variety of focal points, including the river, the barge, the people on the bridge, and so on. However, when a duck takes flight, it captures the attention of everyone present. How is it that the brain possesses such remarkable capabilities in processing such biological objects? This book aims to shed light on this question by demonstrating that brains are particularly adept at leveraging the geometric properties inherent in these objects.

Central to the geometric approach is the concept of a manifold, which expands the notion of a surface to multiple dimensions. The manifold can be defined through collections of n-dimensional data points or the trajectories of a system traversing state space. Similar to how tangent planes are employed to analyze the local linear behavior of points on a surface, the extension to tangent spaces enables the investigation of the local linear behavior of manifolds. The majority of the geometric techniques introduced in this book revolve around manipulating tangent spaces.

By applying these techniques, researchers have been able to analyze various aspects of neuroscience, such as color and spatial vision measurements, as well as the control of eye and arm movements. Furthermore, additional examples are provided to showcase the broader applications of the geometric approach and demonstrate its potential to yield new techniques for exploring neural systems.

One of the significant advantages of adopting a geometric approach is its ability to facilitate visual illustration of the concepts. Each example is accompanied by comprehensive diagrams that effectively complement the descriptions, enhancing the reader's understanding and comprehension. This book is designed with a readership.

Weight: 430g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030984946
Edition number: 1st ed. 2022